ThotDjehuty
d780ed81d7
Adds 9 new top-level / sub-modules to the Rust API only (no Python
bindings yet), with at least one analytic unit test per module.
New Rust modules:
- optimal_control::matrix_riccati (RK4 backward solver)
- timeseries_utils::nonsync_covariance (Hayashi-Yoshida)
- timeseries_utils::wavelet (Haar / Daubechies DWT and MODWT)
- risk_measures (VaR, CVaR, projected sub-gradient CVaR minimisation)
- graph::laplacian + graph::spectral_clustering (Jacobi + k-means++)
- topology (Vietoris-Rips persistent homology, bottleneck distance)
- volterra (Caputo Adams, Markovian lift, second-kind Volterra,
Fourier inversion of characteristic functions)
- signatures (truncated tensor signature, log-sig, random reservoir,
Salvi-Cass-Lyons signature kernel, shuffle product)
All previously stable APIs untouched; abi3-py38 ABI preserved.
New module tests: 29/29 passing. Pre-existing 5 unrelated failures
unchanged.
68 lines
2.0 KiB
ReStructuredText
68 lines
2.0 KiB
ReStructuredText
Topological Data Analysis
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=========================
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The module :code:`topology` implements Vietoris--Rips persistent
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homology and the bottleneck distance between persistence diagrams.
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Vietoris--Rips Filtration
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-------------------------
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For a finite point cloud :math:`X = \{x_1, \dots, x_n\} \subset \mathbb{R}^d`
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and scale :math:`\varepsilon \ge 0`, the Vietoris--Rips complex is
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.. math::
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\mathrm{VR}_\varepsilon(X)
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\;=\;
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\big\{ \sigma \subseteq X : \mathrm{diam}(\sigma) \le \varepsilon \big\}.
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Increasing :math:`\varepsilon` yields a filtration; the persistent
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homology of this filtration produces, for each homological degree
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:math:`k`, a multiset of birth/death pairs
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.. math::
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D_k(X) = \big\{ (b_i, d_i) : 0 \le b_i < d_i \le \infty \big\}.
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Persistence Algorithm
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---------------------
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The boundary matrix :math:`\partial` is built over :math:`\mathbb{Z}/2`
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and reduced left-to-right: for each column :math:`j` we cancel its
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lowest entry by adding any earlier column with the same low. Pairs
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:math:`(\mathrm{low}(j), j)` give birth/death pairs.
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Bottleneck Distance
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-------------------
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For two diagrams :math:`D` and :math:`D'`,
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.. math::
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d_B(D, D')
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\;=\;
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\inf_{\eta : D \to D'}\;
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\sup_{x \in D}\, \|x - \eta(x)\|_\infty,
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where matchings may pair points with the diagonal
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:math:`\Delta = \{(t, t) : t \ge 0\}` at cost :math:`(d - b)/2`.
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The implementation binary-searches the threshold :math:`\varepsilon`
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and certifies a perfect matching by Hopcroft--Karp on the bipartite
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graph of admissible edges.
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API
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---
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.. code-block:: rust
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pub struct PersistencePair { pub dim: usize, pub birth: f64, pub death: f64 }
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pub struct PersistenceDiagram { pub pairs: Vec<PersistencePair> }
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pub fn vietoris_rips_filtration(points: &[Vec<f64>], max_dim: usize, max_eps: f64)
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-> Result<Vec<Simplex>>;
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pub fn persistent_homology(points: &[Vec<f64>], max_dim: usize, max_eps: f64)
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-> Result<PersistenceDiagram>;
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pub fn bottleneck_distance(d1: &[PersistencePair], d2: &[PersistencePair])
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-> Result<f64>;
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